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Theorem bnj927 32114
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj927.1  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj927.2  |-  C  e. 
_V
Assertion
Ref Expression
bnj927  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p )

Proof of Theorem bnj927
StepHypRef Expression
1 simpr 461 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  f  Fn  n )
2 vex 3081 . . . . . 6  |-  n  e. 
_V
3 bnj927.2 . . . . . 6  |-  C  e. 
_V
42, 3fnsn 5582 . . . . 5  |-  { <. n ,  C >. }  Fn  { n }
54a1i 11 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  { <. n ,  C >. }  Fn  {
n } )
6 bnj521 32080 . . . . 5  |-  ( n  i^i  { n }
)  =  (/)
76a1i 11 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( n  i^i  { n } )  =  (/) )
8 fnun 5628 . . . 4  |-  ( ( ( f  Fn  n  /\  { <. n ,  C >. }  Fn  { n } )  /\  (
n  i^i  { n } )  =  (/) )  ->  ( f  u. 
{ <. n ,  C >. } )  Fn  (
n  u.  { n } ) )
91, 5, 7, 8syl21anc 1218 . . 3  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( f  u.  { <. n ,  C >. } )  Fn  (
n  u.  { n } ) )
10 bnj927.1 . . . 4  |-  G  =  ( f  u.  { <. n ,  C >. } )
1110fneq1i 5616 . . 3  |-  ( G  Fn  ( n  u. 
{ n } )  <-> 
( f  u.  { <. n ,  C >. } )  Fn  ( n  u.  { n }
) )
129, 11sylibr 212 . 2  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  ( n  u.  { n } ) )
13 df-suc 4836 . . . . . 6  |-  suc  n  =  ( n  u. 
{ n } )
1413eqeq2i 2472 . . . . 5  |-  ( p  =  suc  n  <->  p  =  ( n  u.  { n } ) )
1514biimpi 194 . . . 4  |-  ( p  =  suc  n  ->  p  =  ( n  u.  { n } ) )
1615adantr 465 . . 3  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  p  =  ( n  u.  { n } ) )
1716fneq2d 5613 . 2  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( G  Fn  p  <->  G  Fn  (
n  u.  { n } ) ) )
1812, 17mpbird 232 1  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    u. cun 3437    i^i cin 3438   (/)c0 3748   {csn 3988   <.cop 3994   suc csuc 4832    Fn wfn 5524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-reg 7921
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-id 4747  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-fun 5531  df-fn 5532
This theorem is referenced by:  bnj941  32118  bnj929  32281
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